Nonviolation of the CHSH inequality under local spin-1 measurements on two spin qutrits

Abstract

In the present paper, based on the general analytical expression [arXiv:2412.03470] for the maximum of the CHSH expectation under local Alice and Bob spin-$s$ measurements in a two-qudit state of dimension $d=2s+1$, $s\geq 1/2$, we analyze whether or not, under spin-$1$ measurements in an arbitrary two-qutrit state, the CHSH inequality is violated. We find analytically for a variety of pure nonseparable two-qutrit states and also, numerically for $1,000,000$ randomly generated pure nonseparable two-qutrit states, that, under local Alice and Bob spin-$1$ measurements in each of these nonseparable states, including maximally entangled, the CHSH inequality is not violated. These results together with the spectral decomposition of a mixed state lead us to the Conjecture that, under local Alice and Bob spin-$1$ measurements, every nonseparable two-qutrit state, pure or mixed, does not violate the CHSH inequality. For a variety of pure two-qutrit states, we further find the values of their concurrence and compare them with the values of their spin-$1$ CHSH parameter, which determines violation or nonviolation by a two-qutrit state of the CHSH inequality under spin-$1$ measurements. This comparison indicates that, in contrast to spin-$\frac{1}{2}$ measurements, where the spin-$\frac{1}{2}$ CHSH parameter of a pure two-qubit state is increasing monotonically with a growth of its entanglement, for a pure two-qutrit state, this is not the case. In particular, for the two-qutrit GHZ state, which is maximally entangled, the spin-$1$ CHSH parameter is equal to $\sqrt{\frac{8}{9}}$, while, for some separable pure two-qutrit states, this parameter can be equal to unity. Moreover, for the two-qutrit Horodecki state, the spin-$1$ CHSH parameter is equal to $4\sqrt{2}/21<1$ regardless of the entanglement type of this mixed state.